For a normal curve with μ = 56 and σ = 10, how much area will be found under the curve to the left of the value 56?
Understand the Problem
The question is asking for the area under a normal distribution curve to the left of the value 56, given a mean (μ) of 56 and a standard deviation (σ) of 10. Since 56 is the mean, we are looking for the cumulative area to the left of the mean in a normal distribution.
Answer
The area under the curve to the left of the value 56 is $0.5$.
Answer for screen readers
The area under the curve to the left of the value 56 is $0.5$.
Steps to Solve
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Identify the Mean and Standard Deviation The mean ($\mu$) of the normal distribution is given as 56, and the standard deviation ($\sigma$) is given as 10.
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Understanding the Normal Distribution In a normal distribution, the area under the curve to the left of the mean (which is also the median) is always 0.5 (or 50%) because it is symmetrical.
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Result Thus, since we are looking for the area to the left of the mean (56), the area is calculated as: $$ P(X \leq 56) = 0.5 $$
The area under the curve to the left of the value 56 is $0.5$.
More Information
In a normal distribution, the mean divides the curve into two equal parts. Therefore, half of the area (50%) lies to the left of the mean, regardless of the specific values of mean and standard deviation.
Tips
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